#3534 ⟨
a
,
b
,
c
|
bb
=
ac
,
bca
=
b
⟩
Up:
Monoid enumeration
Prev:
#3533
⟨
a
,
b
,
c
|
bb
=
ac
,
bca
=
a
⟩
Next:
#3535
⟨
a
,
b
,
c
|
bb
=
ac
,
bca
=
c
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
d
is not injective:
d
⋅ (
da
)
2
=
d
and
d
⋅ 1 =
d
, however (
da
)
2
≠ 1
Rewriting system
Format:
Pretty
Plain
Word:
Enter a word above to compute its normal form. Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Strategy:
Leftmost
Rightmost
Result:
1
1
Auxiliary generators:
d
=
bc
Reduction order:
Left-to-right recursive path with deg(
a
) = deg(
d
) = 0,
a
<
d
; deg(
b
) = 1; deg(
c
) = 2
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
d
(
da
)
2
⇒
d
[5]
2.
b
⇒
da
[4]
3.
ac
⇒ (
da
)
2
[6]
4.
dc
⇒
d
2
ad
[7]
# abc:bb=ac,bca=b ad/b/c bc=d morph:2/1 ddada=d b=da ac=dada dc=ddad